GATE CS 2015 Set 1 — Question 43
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Digital Logic → Boolean Algebra & Logic Gates → Boolean Laws & Theorems
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Question
Consider the operations and .Which one of the following is correct?
Correct answer
(B) Only \f\ is functionally complete
Solution
A set of functions is functionally complete if it can generate all possible boolean functions (or equivalently, if it can generate NOT and AND/OR).Analyze :Since , it is simply a projection (identity) function. It cannot generate NOT or any other operation. Thus, is not functionally complete.Analyze :Check Post's Functional Completeness Theorem criteria (T0, T1, Self-Dual, Monotonic, Linear):
1.T0 (Preserves 0): . Not T0.
2.T1 (Preserves 1): . Not T1.
Since is neither T0 nor T1, it can generate a NOT function (e.g., or similar). With NOT and the complex structure allowing AND/OR generation (it is not linear or monotonic), it is functionally complete.Therefore, only is functionally complete.Continue learning with Success Tracker
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